Abstract

Recently, Samet et al. (2012) introduced the notion of --contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weak --contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize the results of Samet et al.

1. Introduction

The notion of partial metric is one of the most useful and interesting generalizations of the classical concept of metric. The partial metric spaces were introduced in 1994 by Matthews [1] as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial metric context for applications in program verification. Later on, many authors studied the existence of several connections between partial metrics and topological aspects of domain theory (see [2–8] and the references therein). On the other hand, some researchers [9, 10] investigated the characterization of partial metric 0-completeness in terms of fixed point theory, extending the characterization of metric completeness [11–14].

Recently, Samet et al. [15] introduced the notion of --contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weak --contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize Theorems 2.1–2.3 of [15] and many others. An application to ordinary differential equations concludes the paper.

2. Preliminaries

In this section, we recall some definitions and some properties of partial metric spaces that can be found in [1, 5, 10, 16, 17]. A partial metric on a nonempty set is a function such that, for all , we have), (), (), ().

A partial metric space is a pair such that is a non-empty set and is a partial metric on . It is clear that if , then from and it follows that . But if , may not be . A basic example of a partial metric space is the pair , where for all . Other examples of partial metric spaces which are interesting from a computational point of view can be found in [1].

Each partial metric on generates a topology on which has as a base the family of open -balls , where
for all and .

Let be a partial metric space. A sequence in converges to a point if and only if .

A sequence in is called a Cauchy sequence if there exists (and is finite)

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that

A sequence in is called 0-Cauchy if . We say that is 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that .

On the other hand, the partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an example of a 0-complete partial metric space which is not complete.

It is easy to see that every closed subset of a complete partial metric space is complete.

Notice that if is a partial metric on , then the function given by
is a metric on . Furthermore, if and only if

Lemma 1 (see [1, 16]). Let be a partial metric space. Then (a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space , (b)a partial metric space is complete if and only if the metric space is complete.

Let be a non-empty set. If is a partial metric space and is a partially ordered set, then is called an ordered partial metric space. Then are called comparable if or holds. Let be a partially ordered set, and let be a mapping. is a non-decreasing mapping if whenever for all .

Example 3. Let , and define the function by
Then, every non-decreasing mapping is -admissible. For example the mappings defined by and for all are -admissible.

3. Main Results

Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. We start the main section by presenting the new notion of weak --contractive mappings.

Denote by the family of non-decreasing functions such that and for each , where is the th iterate of .

Remark 4. Notice that the family used in this paper is larger (less restrictive) than the corresponding family of functions defined in [15], see also next Examples 12–13.

Lemma 5. For every function , one has for each .

Definition 6. Let be a partial metric space, and let be a given mapping. We say that is a weak --contractive mapping if there exist two functions and such that
for all . If
for all , then is an --contractive mapping.

Remark 7. If satisfies the contraction mapping principle, then is a weak --contractive mapping, where for all and for all and some .

In the sequel, we consider the following property of regularity. Let be a partial metric space, and let be a function. Then(r) is -regular if for each sequence , such that for all and , we have that for all , (c) has the property (C) with respect to if for each sequence , such that for all , there exists such that for all .

Remark 8. Let be a non-empty set, and let be a function. Denote
If is a transitive relation on , then has the property (C) with respect to .

In fact, if is a sequence such that for all , then for all . Now, fix and show that

Obviously, (9) holds if . Assume that (9) holds for some . From , since is transitive, we get . This implies that , and so for all ; that is, has the property (C) with respect to .

Remark 9. Let be an ordered partial metric space, and let be a function defined by
Then has the property (C) with respect to . Moreover, if for each sequence , such that for all convergent to some , we have for all , and then is -regular.

By Remark 8, has the property (C) with respect to . Now, let be a sequence such that for all convergent to some , and then for all , and hence for all . This implies that for all , and so is -regular.

Our first result is the following theorem that generalizes Theorem 2.1 of [15].

Theorem 10. Let be a complete partial metric space, and let be a weak --contractive mapping satisfying the following conditions: (i) is -admissible,(ii)there exists such that , (iii) has the property (C) with respect to ,(iv) is continuous on .Then, has a fixed point, that is; there exists such that .

Proof. Let such that . Define the sequence in by
If for some , then is a fixed point for . Assume that for all . Since is -admissible, we have
By induction, we get
Applying inequality (6) with and and using (13), we obtain
If , from
we obtain a contradiction; therefore, , and hence
By induction, we get
This implies that
Fix , and let such that
Since has the property (C) with respect to , there exists such that for all . Let with , and we show that
Note that (20) holds for . Assume that (20) holds for some , then
This implies that (20) holds for , and hence
Thus, we proved that is a Cauchy sequence in the partial metric space and hence, by Lemma 1, in the metric space . Since is complete, by Lemma 1, also is complete. This implies that there exists such that as ; that is,
From the continuity of on , it follows that as . By the uniqueness of the limit, we get ; that is, is a fixed point of .

In the next theorem, which is a proper generalization of Theorem 2.2 in [15], we omit the continuity hypothesis of . Moreover, we assume -completeness of the space.

Theorem 11. Let be a -complete partial metric space, and let be a weak --contractive mapping satisfying the following conditions: (i) is -admissible,(ii)there exists such that ,(iii) has the property (C) with respect to ,(iv) is -regular.Then, has a fixed point.

Proof. Let such that . Define the sequence in by , for all . Following the proof of Theorem 10, we know that for all and that is a -Cauchy sequence in the -complete partial metric space . Consequently, there exists such that
On the other hand, from for all and the hypothesis (iv), we have
Now, using the triangular inequality, (6) and (25), we get
Since as , for great enough, we have
and hence
This is a contradiction, and so we obtain ; that is, .

Example 12. Let and be defined by for all . Clearly, is a complete partial metric space. Define the mapping by
At first, we observe that we cannot find such that
for all , since we have
for all . Now, we define the function by
Clearly is a weak --contractive mapping with for all . In fact, for all , we have
Moreover, there exists such that . In fact, for , we have
Obviously, is continuous on since , and so we have to show that is -admissible. In doing so, let such that . This implies that , and by the definitions of and , we have
Then, is -admissible. Moreover, if is a sequence such that for all , then for all , and hence for all . Thus, has the property (C) with respect to .Now, all the hypotheses of Theorem 10 are satisfied, and so has a fixed point. Notice that Theorem 10 (also Theorem 11) guarantees only the existence of a fixed point but not the uniqueness. In this example, and are two fixed points of .Moreover, , and so is not an --contractive mapping in the sense of [15] with respect to the complete metric space ; that is, Theorem 2.1 of [15] cannot be applied in this case.

Now, we give an example involving a mapping that is not continuous. Also, this example shows that our Theorem 11 is a proper generalization of Theorem 2.2 in [15].

Example 13. Let and as in Example 12. Clearly, is a -complete partial metric space which is not complete. Then, Theorem 10 is not applicable in this case. Define the mapping by
It is clear that is not continuous at with respect to the metric . Define the function by
Clearly is a weak --contractive mapping with for all . In fact, for all , we have
Proceeding as in Example 12, the reader can show that all the required hypotheses of Theorem 11 are satisfied, and so has a fixed point. Here, and are two fixed points of .

Moreover, since is not complete, where for all , we conclude that neither Theorem 2.1 nor Theorem 2.2 of [15] can be applied to cover this case, also because .

To ensure the uniqueness of the fixed point, we will consider the following hypothesis: (H)for all with , there exists such that , , and .

Theorem 14. Adding condition to the hypotheses of Theorem 10 (resp., Theorem 11), one obtain the uniqueness of the fixed point of .

Proof. Suppose that and are two fixed points of with . If , using (6), we get
which is a contradiction, and so . If by (H), there exists such that
Since is -admissible, from (40), we get
Let for all . Using (41) and (6), we have
Now, let . If is an infinite subset of , then
Then, letting with in the previous inequality, we get
If is a finite subset of , then there exists such that
This implies that
Then, letting , we get
Similarly, using (41) and (6), we get
Since , using (47) and (48), we deduce that
Now, the uniqueness of the limit gives us . This finishes the proof.

Corollary 15. Let be a complete partial metric space, and let be an --contractive mapping satisfying the following conditions: (i) is -admissible, (ii)there exists such that , (iii) has the property with respect to , (iv) is continuous on .Then, has a fixed point.

Corollary 16. Let be a -complete partial metric space, and let be an --contractive mapping satisfying the following conditions: (i) is -admissible,(ii)there exists such that ,(iii) has the property with respect to ,(iv) is -regular.Then, has a fixed point.

Corollary 17. One adds to the hypotheses of Corollary 15 (resp., Corollary 16) the following condition: (HC) for all with , there exists such that and , and one obtains the uniqueness of the fixed point of .

4. Consequences

Now, we show that many existing results in the literature can be deduced easily from our theorems.

4.1. Contraction Mapping Principle

Theorem 18 (Matthews [1]). Let be a -complete partial metric space, and let be a given mapping satisfying
for all , where . Then has a unique fixed point.

Proof. Let be defined by , for all , and let be defined by . Then is an --contractive mapping. It is easy to show that all the hypotheses of Corollaries 16 and 17 are satisfied. Consequently, has a unique fixed point.

Remark 19. In Example 12, Theorem 18 cannot be applied since . However, using our Corollary 15, we obtain the existence of a fixed point of .

4.2. Fixed Point Results in Ordered Metric Spaces

The existence of fixed points in partially ordered sets has been considered in [18]. Later on, some generalizations of [18] are given in [19–24]. Several applications of these results to matrix equations are presented in [18]; some applications to periodic boundary value problems and particular problems are given in [22, 23], respectively.

In this section, we will show that many fixed point results in ordered metric spaces can be deduced easily from our presented theorems.

4.2.1. Ran and Reurings Type Fixed Point Theorem

In 2004, Ran and Reurings proved the following theorem.

Theorem 20 (Ran and Reurings [18]). Let be a partially ordered set, and suppose that there exists a metric in such that the metric space is complete. Let be a continuous and non-decreasing mapping with respect to . Suppose that the following two assertions hold: (i)there exists such that for all with ,(ii)there exists such that ,(iii) is continuous. Then, has a fixed point.

From Theorem 10, we deduce the following generalization and extension of the Ran and Reurings theorem in the framework of ordered complete partial metric spaces.

Theorem 21. Let be an ordered complete partial metric space, and let be a non-decreasing mapping with respect to . Suppose that the following assertions hold: (i)there exists such that for all with ,(ii)there exists such that ,(iii) is continuous on .Then, has a fixed point.

Proof. Define the function by
From (i), we have
Then, is a weak --contractive mapping. Now, let such that . By the definition of , this implies that . Since is a non-decreasing mapping with respect to , we have , which gives us that . Then is -admissible. From (ii), there exists such that , and so . Moreover, by Remark 9, has the property (C) with respect to .Therefore, all the hypotheses of Theorem 10 are satisfied, and so has a fixed point.

Example 22. Let and be defined by for all . Clearly, is a complete partial metric space. Define the mapping by
Clearly is a continuous mapping with respect to the metric . We endow with the usual order of real numbers. Now, condition of Theorem 21 is not satisfied for . In fact, if we assume the contrary, then
which is a contradiction. Then, we cannot apply Theorem 21 to prove the existence of a fixed point of .

Define the function by
It is clear that
Then, is a weak --contractive mapping with for all . Now, let such that . By the definition of , this implies that . Then we have , and so is -admissible. Also, for , we have . Consequently, all the hypotheses of Theorem 10 are satisfied, then we deduce the existence of a fixed point of . Here is a fixed point of .

4.2.2. Nieto and Rodríguez-López Type Fixed Point Theorem

In 2005, Nieto and Rodríguez-López proved the following theorem.

Theorem 23 (Nieto and Rodríguez-López [22]). Let be a partially ordered set, and suppose that there exists a metric in such that the metric space is complete. Let be a non-decreasing mapping with respect to . Suppose that the following assertions hold: (i)there exists such that for all with ,(ii)there exists such that ,(iii)if is a non-decreasing sequence in such that as , then for all .Then, has a fixed point.

From Theorem 11, we deduce the following generalization and extension of the Nieto and Rodríguez-López theorem in the framework of ordered -complete partial metric spaces.

Theorem 24. Let be an ordered -complete partial metric space, and let be a non-decreasing mapping with respect to . Suppose that the following assertions hold: (i)there exists such that for all with , (ii)there exists such that , (iii)if is a non-decreasing sequence in such that as , then for all . Then, has a fixed point.

Proof. Define the function by
The reader can show easily that is a weak --contractive and -admissible mapping. Now, by Remark 9, has the property (C) with respect to and is -regular. Thus all the hypotheses of Theorem 11 are satisfied, and has a fixed point.

Remark 25. In, Example 22, also Theorem 24 cannot be applied since condition is not satisfied.

Remark 26. To establish the uniqueness of the fixed point, Ran and Reurings, Nieto and Rodríguez-López [18, 22] considered the following hypothesis: (u) for all , there exists such that and . Notice that in establishing the uniqueness it is enough to assume that (u) holds for all that are not comparable. This result is also a particular case of Corollary 17. Precisely, if are not comparable, then there exists such that and . This implies that and , and here, we consider the same function used in the previous proof. Then, hypothesis of Corollary 17 is satisfied, and so we deduce the uniqueness of the fixed point. For establishing the uniqueness of the fixed point in Theorems 21 and 24, we consider the following hypothesis: (U) for all that are not comparable, there exists such that , , and .

5. Application to Ordinary Differential Equations

In this section, we present a typical application of fixed point results to ordinary differential equations. In fact, in the literature there are many papers focusing on the solution of differential problems approached via fixed point theory (see, e.g., [15, 25, 26] and the references therein). For such a case, even without any additional problem structure, the optimal strategy can be obtained by finding the fixed point of an operator which satisfies a contractive condition in certain spaces.

Here, we consider the following two-point boundary value problem for second order differential equation:
where is a continuous function. Recall that the Green's function associated to (58) is given by

Let () be the space of all continuous functions defined on . It is well known that such a space with the metric given by
is a complete metric space.

Now, we consider the following conditions:(i)for all , for all with , we have
where , (ii)there exists such that , (iii)for all ,

Theorem 27. Suppose that conditions – hold. Then (58) has at least one solution .

Proof. Consider endowed with the partial metric given by
where . It is easy to show that is -complete but is not complete. In fact,
and consequently is not complete.On the other hand, it is well known that , and is a solution of (58), is equivalent to is a solution of the integral equation
Define the operator by
Then solving problem (58) is equivalent to finding that is a fixed point of . Now, let such that . From (i), we have
Note that for all , , which implies that
Then, for all such that , we have
Define the function by
For all , we have
Then, is an --contractive mapping. From condition (iii), for all , we get
Then, is -admissible. From conditions (ii) and (iii), there exists such that . Thus, all the conditions of Corollary 16 are satisfied, and hence we deduce the existence of such that ; that is, is a solution to (58).

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under Grant no. NRU56000508).

References

S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728 of Annals of the New York Academy of Sciences, pp. 183–197, 1994.

S. J. O'Neill, “Partial metrics, valuations and domain theory,” in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 806 of Annals of the New York Academy of Sciences, pp. 304–315, 1996.